CompMath seminar - Ingo Steldermann, RWTH Aachen University

Title: Boris Galërkin and Barré de Saint-Venant meet for a cup of tea

Abstract:

The Saint-Venant equations or shallow water equations (SWEs) in multiple-dimensions are successfully applied to simulate geophysical flows like river floods, tsunamis, sediment transport, or debris flows. While depth-averaged models like the SWEs are computationally very attractive, information on the vertical velocity is required a priori, typically by assuming a parametrized profile using a time-independent function, e.g. a constant.

Relaxating this limitation seems to be a recurring research subject in the geophysical community and is the motivation for models

like the shear shallow flow model, vertically averaged momentum equations and shallow moment method. All methods have in

common that they can be understood as Galerkin projections of three-dimensional flow models.

The focus of our work is the shallow moment method. The method retains transient information about the vertical flow profile by using a finite Legendre expansion to approximate said vertical velocity with time- and space dependent coefficients. Applying Galerkin projections and choosing an increasing number of basis functions allows to generate a hierarchy of models that in the limit, recover the reference equations of the vertically fully resolved model. However, of practial interest is the usage of a small number of basis functions. The claim is that already for a few basis function significantly increase the model’s predictive power compared to the classical SWEs.

In the talk, I will introduce the shallow moment and shear shallow flow method as possible examples to move byond the SWEs while staying within a depth-averaged framework. Afterwards, I want to focus on one geophysical application and discuss my work on the problem of choosing appropriate friction-models. This is a interesting questions since there are two ways to look at it: top-down from the viewpoint of the Navier-Stokes equations and bottom-up from the viewpoint of the SWEs.